Dirichlet type extensions of Euler sums
In this paper, we study the alternating Euler $T$-sums and $\tilde{S}$-sums, which are infinite series involving (alternating) odd harmonic numbers, and have similar forms and close relations to the Dirichlet beta functions. By using the method of residue computations, we establish the explicit form...
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Académie des sciences
2023-09-01
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| Series: | Comptes Rendus. Mathématique |
| Online Access: | https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.453/ |
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| author | Xu, Ce Wang, Weiping |
| author_facet | Xu, Ce Wang, Weiping |
| author_sort | Xu, Ce |
| collection | DOAJ |
| description | In this paper, we study the alternating Euler $T$-sums and $\tilde{S}$-sums, which are infinite series involving (alternating) odd harmonic numbers, and have similar forms and close relations to the Dirichlet beta functions. By using the method of residue computations, we establish the explicit formulas for the (alternating) linear and quadratic Euler $T$-sums and $\tilde{S}$-sums, from which, the parity theorems of Hoffman’s double and triple $t$-values and Kaneko–Tsumura’s double and triple $T$-values are further obtained. As supplements, we also show that the linear $T$-sums and $\tilde{S}$-sums are expressible in terms of colored multiple zeta values. Some interesting consequences and illustrative examples are presented. |
| format | Article |
| id | doaj-art-99b16ab83b2c4ee58533f05c56bba5eb |
| institution | Kabale University |
| issn | 1778-3569 |
| language | English |
| publishDate | 2023-09-01 |
| publisher | Académie des sciences |
| record_format | Article |
| series | Comptes Rendus. Mathématique |
| spelling | doaj-art-99b16ab83b2c4ee58533f05c56bba5eb2025-02-07T11:09:17ZengAcadémie des sciencesComptes Rendus. Mathématique1778-35692023-09-01361G6979101010.5802/crmath.45310.5802/crmath.453Dirichlet type extensions of Euler sumsXu, Ce0Wang, Weiping1School of Mathematics and Statistics, Anhui Normal University, Wuhu 241002, P.R. ChinaSchool of Science, Zhejiang Sci-Tech University, Hangzhou 310018, P.R. ChinaIn this paper, we study the alternating Euler $T$-sums and $\tilde{S}$-sums, which are infinite series involving (alternating) odd harmonic numbers, and have similar forms and close relations to the Dirichlet beta functions. By using the method of residue computations, we establish the explicit formulas for the (alternating) linear and quadratic Euler $T$-sums and $\tilde{S}$-sums, from which, the parity theorems of Hoffman’s double and triple $t$-values and Kaneko–Tsumura’s double and triple $T$-values are further obtained. As supplements, we also show that the linear $T$-sums and $\tilde{S}$-sums are expressible in terms of colored multiple zeta values. Some interesting consequences and illustrative examples are presented.https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.453/ |
| spellingShingle | Xu, Ce Wang, Weiping Dirichlet type extensions of Euler sums Comptes Rendus. Mathématique |
| title | Dirichlet type extensions of Euler sums |
| title_full | Dirichlet type extensions of Euler sums |
| title_fullStr | Dirichlet type extensions of Euler sums |
| title_full_unstemmed | Dirichlet type extensions of Euler sums |
| title_short | Dirichlet type extensions of Euler sums |
| title_sort | dirichlet type extensions of euler sums |
| url | https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.453/ |
| work_keys_str_mv | AT xuce dirichlettypeextensionsofeulersums AT wangweiping dirichlettypeextensionsofeulersums |